On the reduced space of multiplicative multivectors

A strict Lie $2$-algebra $\Gamma(\wedge^\bullet A) \stackrel{T}{\rightarrow} \mathfrak{X}{$\mathrm{m}\mathrm{u}\mathrm{l}\mathrm{t}$}^\bullet($\mathcal{G}$)$ is associated with any Lie groupoid $\mathcal{G}$. Here, $\mathrm{\Gamma }({\wedge }^{\bullet }A)$ is the Schouten algebra of the tangent Lie algebroid $A$ of $\mathcal{G}$ and ${\mathfrak{X}}_{\mathrm{m}\mathrm{u}\mathrm{l}\mathrm{t}}^{\bullet }(\mathcal{G})$ is the space of multiplicative multivectors on $\mathcal{G}$. The quotient ${R}{$\mathrm{m}\mathrm{u}\mathrm{l}\mathrm{t}$}^\bullet:=\mathfrak{X}_{$\mathrm{m}\mathrm{u}\mathrm{l}\mathrm{t}$}^\bullet($\mathcal{G}$)/$\mathrm{I}\mathrm{m}\mathrm{g}$ T$,aMoritainvariantof$$\mathcal{G}$$, is called the reduced space of multiplicative multivectors. We prove a canonical decomposition formula of elements in ${\mathfrak{X}}_{\mathrm{m}\mathrm{u}\mathrm{l}\mathrm{t}}^{\bullet }(\mathcal{G})$ and establish a key relation between $R_{\mathrm{m}\mathrm{u}\mathrm{l}\mathrm{t}}^k$ and the cohomology $$\mathrm{H}$ ^1(\mathfrak{J} $\mathcal{G}$,\wedge^k A)$where$\mathfrak{J} $\mathcal{G}$$ is the jet groupoid of $\mathcal{G}$ and $1⩽k⩽\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{k}A$. We also study $R_{\mathrm{d}\mathrm{i}\mathrm{f}\mathrm{f}}^{\bullet }$, the reduced space of Lie algebroid differentials on $A$. By taking infinitesimals, $\bar{\delta }:$ $R_{\mathrm{m}\mathrm{u}\mathrm{l}\mathrm{t}}^{\bullet }$ $\to$ $R_{\mathrm{d}\mathrm{i}\mathrm{f}\mathrm{f}}^{\bullet }$, the two reduced spaces are related. We find that the kernel of $\bar{\delta }$ is isomorphic to the kernel of the Van Est map $$\mathrm{H}$^1($\mathcal{G}$,\wedge^k \ker\rho)\to $\mathrm{H}$^1(A,\wedge^k \ker\rho)$,where$\rho$istheanchorof$A$.